1

The DiVincenzo criteria

Five criteria that any candidate quantum computerimplementation must satisfy.

Two additional criteria for quantum communication

IBM Research

2

Implementation of quantum computers

1. Well-defined qubits

2. Initialization to a pure state

3. Universal set of quantum gates

4. Qubit-specific measurement

5. Long coherence times

D. DiVincenzo

|1⟩⟩⟩⟩

|0⟩⟩⟩⟩

10

XOR

R

|00 … 0 ⟩⟩⟩⟩

|0… 0 ⟩⟩⟩⟩ + |1… 1 ⟩⟩⟩⟩

A scalable system with well-defined qubits

The ability to initialize the state of the qubits to a fiducial initial state, such as |00...0⟩

Long relevant decoherence times, muchlonger than the gate operation time

A “universal” set of quantum gates

A qubit specific measurement capability

3

Well-defined qubits

• two-level quantum systems

• two-dimensional subspaces of larger systems

+: auxiliary levels- : leakage

S1/2

P1/2

D3/2397 nm

866 nm

393 nm

854 nm

P3/2

� 1H,13C,19F, ...� electron spin

4

n 2-level systems vs. one 2 n-level system

⊗⊗⊗⊗ ⊗⊗⊗⊗0

1

000

111

001

010

011

100

101

110

or000

111

energy or precision ~ n

000

001 010 100

110101011

111

scalableenergy or precision ~ 2n

NOT scalable

5

Ensemble quantum computer

⊗⊗⊗⊗ ⊗⊗⊗⊗01

⊗⊗⊗⊗ ⊗⊗⊗⊗01

⊗⊗⊗⊗ ⊗⊗⊗⊗01

⊗⊗⊗⊗ ⊗⊗⊗⊗01

⊗⊗⊗⊗ ⊗⊗⊗⊗01

Image: Krenner et al, cond-mat/0505731

13CCl

Cl

Cl

1H

Works fine --only read-out must be modified

Many identical copies of aquantum computer

6

Useless device

01

01

01

01

01

01

01

01

01

01

01

01

01

01

01

Many copies of each qubit,but not a quantum computer

⊗⊗⊗⊗ ⊗⊗⊗⊗

7

Encoded qubits

Beware: leakage

|0⟩⟩⟩⟩L = (|01⟩⟩⟩⟩+|10⟩⟩⟩⟩) |0⟩⟩⟩⟩

|1⟩⟩⟩⟩L = √2/3 |001⟩⟩⟩⟩ + √1/3(|01⟩⟩⟩⟩+|10⟩⟩⟩⟩)|0⟩⟩⟩⟩

|0⟩⟩⟩⟩L = |01⟩⟩⟩⟩ - |10⟩⟩⟩⟩

|1⟩⟩⟩⟩L = |10⟩⟩⟩⟩ + |01⟩⟩⟩⟩Decoherence free subspace

Trade qubits forHamiltonian terms

(e.g. exchange only QC)

8

Initialization to a pure state

Equilibration at low temperature ( hν >> kT )

Other physical mechanisms:

• Ferromagnet• Laser cooling• Optical pumping• . . .

If you want qubits in |000⟩, simply rotate the qubits from |ψ⟩ to |000⟩

To |000⟩⟩⟩⟩

Perform a hard, non-destructive measurementTo |ψψψψ⟩⟩⟩⟩Other physical processes

9

Initialization timescalee.g. equilibration can be slow (> 5 T1)

Why initialization anyways?

Computation = garbage in ⇒ garbage out

Ancilla qubits (“help” qubits)

Error correction = removing entropy from the qubits

one-time

need continuousfresh supply

Either initialize fastor build qubit “conveyer belt”

10

Are mixed states acceptable?

↑↑↑↑↑↑↑↑ ↑↓↑↓↑↓↑↓ ↓↑↓↑↓↑↓↑ ↓↓↓↓↓↓↓↓

Mixed state Effective pure state

Signal same as for pure statebut amplitude ~ 1/2n

↑↑↑↑↑↑↑↑↑↓↑↓↑↓↑↓ ↓↑↓↑↓↑↓↑ ↓↓↓↓↓↓↓↓

Gershenfeld&Chuang, Science 97, Cory, Havel & Fahmi, PNAS 97

Equilibrium at moderate or high temperature ( hν >> kT )

polarization p

11

Effective pure state preparation

(1) Add up 2N-1 experiments (Knill,Chuang,Laflamme, PRA 1998)

Later ≈ (2N - 1) / N experiments (Vandersypen et al., PRL 2000)

(2) Work in subspace (Gershenfeld&Chuang, Science 1997)

(3) Average over space (Cory et al., Phys. D 1998)

↑↑↑↑↑↑↑↑ ↑↓↑↓↑↓↑↓ ↓↑↓↑↓↑↓↑ ↓↓↓↓↓↓↓↓ ↑↑↑↑↑↑↑↑ ↑↓↑↓↑↓↑↓ ↓↑↓↑↓↑↓↑ ↓↓↓↓↓↓↓↓ ↑↑↑↑↑↑↑↑ ↑↓↑↓↑↓↑↓ ↓↑↓↑↓↑↓↑ ↓↓↓↓↓↓↓↓ ↑↑↑↑↑↑↑↑ ↑↓↑↓↑↓↑↓ ↓↑↓↑↓↑↓↑ ↓↓↓↓↓↓↓↓

+ + =

↑↑↑↑↑↑↑↑↑↑↑↑ ↑↑↓↑↑↓↑↑↓↑↑↓ ↑↓↑↑↓↑↑↓↑↑↓↑ ↑↓↓↑↓↓↑↓↓↑↓↓ ↓↑↑↓↑↑↓↑↑↓↑↑ ↓↑↓↓↑↓↓↑↓↓↑↓ ↓↓↑↓↓↑↓↓↑↓↓↑ ↓↓↓↓↓↓↓↓↓↓↓↓ ↑↑↑↑↑↑↑↑↑↑↑↑ ↑↑↑↑↑↓↑↓↑↓↑↓ ↑↑↑↑↓↑↓↑↓↑↓↑ ↑↑↑↑↓↓↓↓↓↓↓↓ ↓↑↑↓↑↑↓↑↑↓↑↑ ↓↑↓↓↑↓↓↑↓↓↑↓ ↓↓↑↓↓↑↓↓↑↓↓↑ ↓↓↓↓↓↓↓↓↓↓↓↓

12

“Scalable” QC with hot qubits

Goal: obtain k cold qubits from n hot qubits

2

Idea: reduce the entropy of k qubits, and increase the entropyof the remaining qubits (total entropy remains constant)

( Pr[0] = p = 1+ε )

Schulman & Vazirani, Proc. 31st ACM Symp Theory of Computing, p.322 (1999)

“Efficient bootstrapping”

Overhead: # qubits n ∼ k

# operations ∼ k log k

( as k and n →→→→ ∞∞∞∞ )

n H(p) = k H(0) + ( n-k ) H(1/2)

k = n (1 - H(p)) ≅ n ε2

13

Building block Schulman-Vazirani cooling

Prob. bc

p2 00

p(1-p) 01(1-p)p 10(1-p)2 11

Prob. bc

p2 00

p(1-p) 01(1-p)p 11(1-p)2 10

CNOT12

If qubit c is 1,qubit b is at ∞∞∞∞ T.

If qubit c is 0,bias qubit b is 2ε.

Fredkinc,ab (swap qubits a and b iff qubit c is 0)

If qubit b is 1, qubit a has bias ε.

If qubit b is 0, qubit a has bias 2ε.

On average, qubit a has bias 3 εεεε/2 , so it has been “cooled”.

Step 1:

Step 2:

14

Universal set of quantum gates| 0| 0| 0| 0 ⟩⟩⟩⟩

| 1| 1| 1| 1 ⟩⟩⟩⟩

xy

zSelective single-qubit rotations

In00011011

Out00011110

Almost any two-qubit gate is universalE.g. quantum-XOR or Controlled-NOT

1 0 0 0

0 1 0 0 0 0 0 1

0 0 1 0

(| 0(| 0(| 0(| 0 ⟩⟩⟩⟩ + | 1| 1| 1| 1 ⟩⟩⟩⟩)√√√√2

| 00| 00| 00| 00 ⟩⟩⟩⟩ + | 11| 11| 11| 11 ⟩⟩⟩⟩

√√√√2

| 00| 00| 00| 00 ⟩⟩⟩⟩ + | 10| 10| 10| 10 ⟩⟩⟩⟩

√√√√2 | 0| 0| 0| 0 ⟩⟩⟩⟩ =

00

01

10

11

00 01 10 11

Sufficient: rotations about two different axis

15

Coupling networks

1 2

3

J12(t)

J23(t)J13(t)

switchable and direct

1 2

3

J12(t)

J23(t)

indirect

1 2 3

bus

J1(t) J2(t) J3(t)

e.g. nearest neighbour

1 2

3

J12

J23J13

fixed

1 2

3

J12

J23

fixed and indirect

1 2 3

bus

J1 J2J3

16

Indirect coupling: Shuffle qubits around

SWAP12 = CNOT12 CNOT21 CNOT12

“only” a linear overhead ...

1 2 3 4

2 1 3 4

2 3 1 4

2 3 1’ 4’

2 1’ 3 4’

1’ 2 3 4’

17

Removing effect of fixed couplings: refocusing

• There exist efficient extensions for arbitrary coupling networksLeung et al, PRA 00, Jones&Knill, JMR 99

• Refocusing can also be used to remove unwanted single-qubit terms

XB180 XB

180

XA180XA

180

18

Even individual addressing is not strictly needed

…

ψψψψ1111 ψψψψ2222 ψψψψ3333 ψψψψn0 1 0 0 0 0 0 0

A1 B1 C1 A2 B2 C2 A3 B3 C3 An Bn Cn

XA flips all A’sCNOTCA selectively flips A2FredkinC_AB swaps A1 and B1etc.

Distinct qubit at the end is needed for setting up a unique “1”

S. Lloyd, Science 261, 1569, 1993

X

19

Other requirements

CalibrationCross-talkLeft-over terms in HamiltonianNon-commuting terms in HamiltonianHardware limitations (pulse timing, phase noise etc)

Gates must be precise (systematic errors)

> 10000 faster than coherence timeParallelization

Gates must be fast

20

Qubit measurement

Ideally: reliable hard measurement of all qubits

Acceptable in principle:

• ensemble averaged measurement of each qubit (next)

• unreliable measurement (next)

• hard measurement of a single qubit(swap consecutive qubits into read-out site,while maintaining coherence)

21

Dealing with a limited measurement fidelity

1) Repeat calculation

OK for “decision problems” (1-bit answer)Not convenient

2) “Quantum FAN-OUT”

⊕⊕

a| 0| 0| 0| 0 ⟩⟩⟩⟩ + b| 1| 1| 1| 1 ⟩⟩⟩⟩

| 0| 0| 0| 0 ⟩⟩⟩⟩

| 0| 0| 0| 0 ⟩⟩⟩⟩

a| 000| 000| 000| 000 ⟩⟩⟩⟩ + b| 111| 111| 111| 111 ⟩⟩⟩⟩ majority vote

1- α

1-βα

β

0

1

‘0'

‘1'

Measurement of one qubit must not disturb state of others, apart from collapse.

22

Ensemble averaged measurements

Say at the end of Shor’s algorithm, we have state

| 0110 ⟩⟩⟩⟩ + | 0011 ⟩⟩⟩⟩

Measurement on a single system gives0100 or 0011

From either outcome, can find prima factor (e.g. “5”) classically

Measurement on ensemble would give0 ½ 1 ½

Instead, perform classical postprocessing on quantum computernow get always “5”, and averaging no longer hurts

23

Decoherence

Decoherence (T2)Relaxation (T1)Leakage

Uncorrelated errors Correlated errors

Random errorsSystematic errors

Over time, or between qubits

The “coherence time” summarizes many aspects of state degradation

detect, and replace by random qubit

detect and correct

unwind

maximum time for measurement

maximum time for computation

24

T1 and T2 (and terminology)

y

z

x

T1

Longitudinal relaxationSpin-lattice relaxationRelaxationEnergy relaxation…

T2

Transverse relaxationSpin-spin relaxationDecoherencePhase randomizationDephasing…

y

x

z

By definition: T2 < 2T1 In practice, often T2 << T1

25

Two additional criteria

6. Interconvert stationary and flying qubits

7. Transmit flying qubits between distant locations

Repeater stationsDistributed quantum computing

Quantum communicationError correction (communication between different parts of the computer)

26

Key challenge

combine access to qubits (initialization, control, readout) with high degree of isolation (coherence)

in a scalable system

Message about DiVincenzorequirements

(almost) everything goesCan trade off one requirement for another